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\large{Name:} \hfill    \large{Probability for Scientists, Fall 2013}

\large{Collaborator(s):} \hfill    \large{ Bio 409 / Bio 509 / Stat 479 } 

\hfill    \large{ Lab 2 (100 pts), Due 10 Sep 2013 }

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\textbf{Please show your work, and leave answers as fractions when applicable.
Approximations are permitted, provided justification is included. Attach extra
paper as needed.}

\section{(20) Craps}
Craps is a betting game where 2 6-sided dice are tossed.  
A toss of 7 or 11 results in an immediate win, and a toss of 2, 3, or
13 results in an immediate loss.  
\begin{enumerate}
    \item What is the probability of neither winning nor losing on a single
toss?
    \item What would this probability be if craps were played with 3 4-sided
dice?
\end{enumerate}
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\section{(20) Blackjack}
Blackjack is a betting game played with a standard deck of cards, where the goal
is to get 21 points.  Each card has
a point value: 4 cards = 11 points (Aces), 16 cards = 10 points (10s and face
cards), and 32 other numbered cards.  
\begin{enumerate}
    \item What is the probability that 2 cards drawn
from a full, well-shuffled deck add to 21?
    \item What proportion of the sample space of 2-card hands scores 20 or more
points?
\end{enumerate}
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\clearpage
\textbf{Probability and information encoding are deeply intertwined topics.  In this
section, we will focus on DNA, which biological systems use 
to encode information.  DNA contains 4 different bases (A, T, G, and C), and is
double-stranded: the 2 strands complement each other (A matches with T
and C matches with G).  Thus, the two strands are completely dependent on each
other, with 4 possible choices at each position.}


\section{(15) Primers}
PCR is a technique used to amplify (make many copies of) DNA.  PCR requires short
stretches of DNA that complement regions of the target DNA.  How many possible unique primers are there
of the following lengths?
\begin{enumerate}
    \item 5
    \item 10
    \item 20
\end{enumerate}
 
\section{(15) Genomes}
The rabies virus has a genome that contains approximately 12,000 bases.  

\begin{enumerate}
    \item How many unique sequences of 12,000 bases are theoretically possible?
    \item Why might all of these sequences not be practically possible?  What
forces constrain genomes?
\end{enumerate}
 
\section{(30) Primers + Genomes}
I give you randomly-selected primers of the following sizes, as well as a
sample of rabies virus.
Assuming the rabies virus genome is random (which it isn't!), what is the
probability that each primer perfectly matches the virus sample at least once?
(Hint: what is the probability that the sequence HTTH is *not* found in 10
random coin flips?)

\begin{enumerate}
    \item 1
    \item 2
    \item 5
    \item 10
    \item 20
    \item 100
\end{enumerate}
 
\end{document}
